Maximum value of `(3sintheta+4costheta)` is:

Answer

To find the maximum value of the expression \(3\sin\theta + 4\cos\theta\), we can use the formula for the maximum value of the function of the form \(a\sin\theta + b\cos\theta\).

### Step-by-Step Solution:

1. **Identify coefficients**:
- In the expression \(3\sin\theta + 4\cos\theta\), we identify \(a = 3\) and \(b = 4\).

2. **Use the formula for maximum value**:
- The maximum value of \(a\sin\theta + b\cos\theta\) is given by the formula:
\[ \sqrt{a^2 + b^2} \]

3. **Calculate \(a^2 + b^2\)**:
- Calculate \(a^2\) and \(b^2\):
\[ a^2 = 3^2 = 9 \]
\[ b^2 = 4^2 = 16 \]
- Now, add these values:
\[ a^2 + b^2 = 9 + 16 = 25 \]

4. **Take the square root**:
- Now, take the square root of \(25\):
\[ \sqrt{25} = 5 \]

5. **Conclusion**:
- Therefore, the maximum value of \(3\sin\theta + 4\cos\theta\) is \(5\).

### Final Answer:
The maximum value of \(3\sin\theta + 4\cos\theta\) is \(5\).

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